Test 1

1. a) Which attributes of particles affect their electromagnetic

interaction? b) Explain the definition of electric field vector. c) Explain the definition of electric potential. d) Explain Gausss law for the electric field.

2. Two identical small charged spheres, each having a mass of m =30 g, hang in equilibrium as shown in the included figure. The length of each string is L =15 cm, and the angle each string makes with the vertical direction is u =5. a) Complete the free body diagram and identify all forces exerted on a (single) sphere. b) From the equilibrium condition, find the magnitude of the electrostatic force1 exerted by each sphere on the other sphere. c) Determine the charge of each sphere.

3. An electron2 is released from rest in a uniform electric field, produced by two parallel plates separated by distance L =10cm, as shown in the figure. The strength of the field is E =10 kN/C. a) Mark the electric field lines and the electrostatic force exerted on the electron. Determine both the magnitude of the electric field and the electrostatic force (vector) exerted on the electron. b) What should be the potential difference between the plates to produce the electric field? c) At what speed does the electron reach the other plate?

4. A uniformly charged insulating rod of length L is bent into the shape of a semicircle (of radius t=Lr ). The total charge of the rod is Q. (See the appropriate page). a) Determine the linear charge density of this object. b) Find the electric potential due to the object at the center of its curvature (point O). c) Find the electric field vector produced by the object at the same point.

It is not possible to determine the sign of the charge. As long as the charges of the spheres are like they will repel each other.

u u u L Lx y T WFel(0.4) (0.5) (0.3) - 3 -

a) The magnitude of the electric field is given in the problem CkN10 E = Consistently with indicated coordinate system, from the definition of the electric field vector the force exerted on the electron is i i i E F

N 10 6 . 1

CN10 1 C 10 6 . 1

eE e15 4 19 = = = =

b) Both electric potential and electric field vector describe electric field. They are therefore related. If the electric field is known in certain region (along a line between two points) the potential difference between these two points is opposite the linear integral of the electric field along this line. For simplicity we can consider a straight line from the initial to the final location of the electron kV 1 m 1 . 0CN10 1 EL dx ) E ( V V V4L0 line = = = = = = } } +ds E

c) From the definition of potential and potential energy, the electrostatic work performed on the electron is related to the potential difference between the two plates ( ) ( ) eV V V e U Wel el = = A = A +

Assuming (reasonable) that the contribution of other interactions is insignificant, the final speed can be determined from the work-energy theorem and the definition of kinetic energy of a particle sm10 8 . 1kg 10 11 . 9V 1000 C 10 6 . 1 2meV 2mK 2mK 2v73119 ~

= =A= =

+

x y E F V- V+(0.3) (0.3) (0.3) (0.3) - 4 -

a) The linear charge density () of this object is related to the charge (dq) of the differential fragment of the object and the size of this piece (length dl). For uniformly charged bodies the ratio of differentials can be replaced by the total values of both quantities. LQdldq= =

b) In order to find the electric potential at a certain location we must add electric potential due to all "point charges" in the body. A contribution dV to the electric potential (with the reference point at infinity), due to a differential fragment dl is | =

c) In order to find the electric field we must add (vectorially) electric fields created by all "point charges" in the body. A contribution E d to the electric field, due to a differential fragment dl, can be found from Coulomb's law. | | | | | | | | |t= | |